Connection (affine bundle)

2017-02-28T04:21:44Z

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Let {{math|''Y'' → ''X''}} be an [[affine bundle]] modelled over a vector bundle {{math|{{overline|''Y''}} → ''X''}}. A [[connection (fibred manifold)|connection]] {{math|Γ}} on {{math|''Y'' → ''X''}} is called the '''affine connection''' if it as a section {{math|Γ : ''Y'' → J1''Y''}} of the [[jet bundle]] {{math|J1''Y'' → ''Y''}} of {{math|''Y''}} is an affine bundle morphism over {{math|''X''}}. In particular, this is the case of an [[affine connection]] on the [[tangent bundle]] {{math|T''X''}} of a [[smooth manifold]] {{math|''X''}}.

With respect to affine bundle coordinates {{math|(''xλ'', ''yi'')}} on {{math|''Y''}}, an affine connection {{math|Γ}} on {{math|''Y'' → ''X''}} is given by the [[connection (fibred manifold)|tangent-valued connection form]]

: <math>\begin{align}\Gamma &=dx^\lambda\otimes \left(\partial_\lambda + \Gamma_\lambda^i\partial_i\right)\,, \\ \Gamma_\lambda^i&={{\Gamma_\lambda}^i}_j\left(x^\nu\right) y^j + \sigma_\lambda^i\left(x^\nu\right)\,. \end{align}</math>

An affine bundle is a fiber bundle with a [[affine group|general affine]] [[fiber bundle|structure group]] {{math|GA(''m'', ℝ)}} of affine transformations of its typical fiber {{math|''V''}} of dimension {{math|''m''}}. Therefore, an affine connection is associated to a [[connection (principal bundle)|principal connection]]. It always exists.

For any affine connection {{math|Γ : ''Y'' → J1''Y''}}, the corresponding [[affine bundle|linear derivative]] {{math|{{overline|Γ}} : {{overline|''Y''}} → J1{{overline|''Y''}}}} of an affine morphism {{math|Γ}} defines a unique [[connection (vector bundle)|linear connection]] on a vector bundle {{math|{{overline|''Y''}} → ''X''}}. With respect to linear bundle coordinates {{math|(''xλ'', {{overline|''y''}}''i'')}} on {{math|{{overline|''Y''}}}}, this connection reads

: <math> \overline \Gamma=dx^\lambda\otimes\left(\partial_\lambda +{{\Gamma_\lambda}^i}_j\left(x^\nu\right) \overline y^j\overline\partial_i\right)\,.</math>

Since every vector bundle is an affine bundle, any linear connection on
a vector bundle also is an affine connection.

If {{math|''Y'' → ''X''}} is a vector bundle, both an affine connection {{math|Γ}} and an associated linear connection {{math|{{overline|Γ}}}} are
connections on the same vector bundle {{math|''Y'' → ''X''}}, and their difference is a basic soldering form on
: <math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes\partial_i \,.</math>
Thus, every affine connection on a vector bundle {{math|''Y'' → ''X''}} is a sum of a linear connection and a basic soldering form on {{math|''Y'' → ''X''}}.

It should be noted that, due to the canonical vertical splitting {{math|V''Y'' {{=}} ''Y'' × ''Y''}}, this soldering form is brought into a [[vector-valued differential form|vector-valued form]]
: <math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes e_i </math>
where {{math|''ei''}} is a fiber basis for {{math|''Y''}}.

Given an affine connection {{math|Γ}} on a vector bundle {{math|''Y'' → ''X''}}, let {{math|''R''}} and {{math|{{overline|''R''}}}} be the [[connection (fibred manifold)|curvatures]] of a connection {{math|Γ}} and the associated linear connection {{math|{{overline|Γ}}}}, respectively. It is readily observed that {{math|''R'' {{=}} {{overline|''R''}} + ''T''}}, where

: <math>\begin{align}
T &=\tfrac12 T_{\lambda\mu}^i dx^\lambda\wedge dx^\mu\otimes \partial_i\,, \\
T_{\lambda \mu}^i &= \partial_\lambda\sigma_\mu^i - \partial_\mu\sigma_\lambda^i + \sigma_\lambda^h {{\Gamma_\mu}^i}_h - \sigma_\mu^h {{\Gamma_\lambda}^i}_h\,,
\end{align}</math>

is the [[connection (fibred manifold)|torsion]] of {{math|Γ}} with respect to the basic soldering form {{math|''σ''}}.

In particular, let us consider the tangent bundle {{math|T''X''}} of a manifold {{math|''X''}} coordinated by {{math|(''xμ'', ''ẋμ'')}}. There is the canonical soldering form
:<math>\theta=dx^\mu\otimes \dot\partial_\mu </math>
on {{math|T''X''}} which coincides with the [[tautological one-form]]
:<math>\theta_X=dx^\mu\otimes \partial_\mu</math>
on {{math|''X''}} due to the canonical vertical splitting {{math|VT''X'' {{=}} T''X'' × T''X''}}. Given an arbitrary linear connection {{math|Γ}} on {{math|T''X''}}, the corresponding affine connection

: <math>\begin{align}
A&=\Gamma +\theta\,, \\
A_\lambda^\mu&={{\Gamma_\lambda}^\mu}_\nu \dot x^\nu +\delta^\mu_\lambda\,,
\end{align}</math>

on {{math|T''X''}} is the [[Cartan connection]]. The torsion of the Cartan connection {{math|''A''}} with respect to the soldering form {{math|''θ''}} coincides with the [[torsion tensor|torsion]] of a linear connection {{math|Γ}}, and its curvature is a sum {{math|''R'' + ''T''}} of the curvature and the torsion of {{math|Γ}}.

==See also==
*[[Connection (fibred manifold)]]
*[[Affine connection]]
*[[Connection (vector bundle)]]
*[[Connection (mathematics)]]
*[[Affine gauge theory]]

==References==
* {{cite book|first1=S. |last1=Kobayashi |first2=K. |last2=Nomizu |title=Foundations of Differential Geometry |volume=1–2 |publisher=Wiley-Interscience |date=1996 |ISBN=0-471-15733-3}}
* {{cite book|authorlink=Gennadi Sardanashvily|last=Sardanashvily |first=G. |title=Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory |publisher=Lambert Academic Publishing |date=2013 |ISBN=978-3-659-37815-7 |arxiv=0908.1886}}

[[Category:Differential geometry]]
[[Category:Connection (mathematics)]]

{{differential-geometry-stub}}

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Connection (affine bundle)